# Parallel-in-Space-and-Time Simulation of the Three-Dimensional, Unsteady Navier-Stokes Equations for Incompressible Flow

## TL;DR

This paper explores combining parallel-in-time and spatial parallelization techniques to accelerate 3D incompressible Navier-Stokes simulations, demonstrating improved speedup on high-performance computing systems.

## Contribution

It introduces a space-time parallel scheme using Parareal combined with spatial parallelization for Navier-Stokes equations, enabling further speedup beyond spatial saturation.

## Key findings

- Space-time parallelization achieves additional speedup.
- Method effective on high-core-count systems.
- Numerical experiments confirm scalability.

## Abstract

In this paper we combine the Parareal parallel-in-time method together with spatial parallelization and investigate this space-time parallel scheme by means of solving the three-dimensional incompressible Navier-Stokes equations. Parallelization of time stepping provides a new direction of parallelization and allows to employ additional cores to further speed up simulations after spatial parallelization has saturated. We report on numerical experiments performed on a Cray XE6, simulating a driven cavity flow with and without obstacles. Distributed memory parallelization is used in both space and time, featuring up to 2,048 cores in total. It is confirmed that the space-time-parallel method can provide speedup beyond the saturation of the spatial parallelization.